A person using a ladder supported on vertical wall is 3/4 of the way up the ladder. If the person climbing the ladder has a weight of 980 newtons and the ladder is 4.89 meters long, how far from the wall can the base of the ladder be placed, and not slip? The coefficient of friction between the base of the ladder and the ground is 0.34. Assume that there is no friction between the ladder and the wall and that the ladder is effectively weightless.
Set the moment about the ladder/wall contact point equal to zero. Assume the ladder has no mass. (That is probably not a good assumption, but they did not provide a value). Let the ladder length be L, and the distance of the ladder bottom from the wall be x. The person's weight (980 N)cancels out. 0.34 is the STATIC friction coefficient. The normal force of the ground on the bottom of the ladder is 980 N.
(x/4)*980 + sqrt(L^2 -x^2)*0.34*980 = x*980
3x/(4L) = 0.34*sqrt[1 - (x/L)^2]
0.5625*(x/L)^2 = 0.1156[1 - (x/L)^2]
0.6781*(x/L)^2 = 0.1156
x/L = 0.413
thanks for your timely help.
To determine how far from the wall the base of the ladder can be placed without slipping, we need to consider the forces acting on the ladder.
First, let's analyze the forces acting on the ladder. The weight of the person, 980 newtons, can be treated as a downward force applied at the center of mass of the person (assumed to be halfway up the ladder). The weight of the ladder is neglected since it is described as effectively weightless.
The vertical component of the weight of the person can be calculated as (3/4) * 980 N = 735 N, acting downward.
Now, let's consider the forces at the base of the ladder. We have the static friction force acting parallel to the ground, opposing the potential sliding motion. The force of static friction (Fs) can be calculated as the product of the coefficient of friction (μ) and the normal force (N).
The normal force (N) can be calculated by summing up all the vertical forces acting at the base, which are the vertical component of the weight of the person and the vertical component of the reaction force exerted by the wall.
N = (3/4) * 980 N + (1/4) * 980 N = 735 N + 245 N = 980 N
Now, we can determine the maximum static friction force:
Fs = μ * N = 0.34 * 980 N = 333.2 N
To prevent the ladder from slipping, the maximum static friction force (333.2 N) must be equal to or greater than the horizontal component of the weight of the person. The horizontal component of the weight can be calculated as (1/4) * 980 N = 245 N.
Since the maximum static friction force (333.2 N) is greater than the horizontal component of the weight (245 N), the base of the ladder can be placed up to a distance where it would not slip.
In this case, the base of the ladder can be placed anywhere within that maximum distance without slipping.